Math Problem Statement
You selected the correct choice, and your second answer is correct, but your first answer is incorrect. Use the derivative of the function to determine whether the function is increasing or decreasing in the intervals around any critical values found in the previous step in order to distinguish minima and maxima. Note that if a function f is increasing on both intervals around a critical value c, it is also increasing at the critical point, even if f prime(c)equals0, and the same holds for decreasing intervals.
Solution
It seems like you're working with a function and its derivative to determine the intervals of increase and decrease around its critical points, as well as distinguishing between minima and maxima. Here's how we can proceed:
Step-by-step approach:
- Find the derivative: Compute of the function .
- Locate the critical points: Critical points occur where or is undefined. You should have already identified these points.
- Determine intervals around critical points: To find out whether the function is increasing or decreasing, examine the sign of the derivative on the intervals that surround each critical point.
- If , the function is increasing.
- If , the function is decreasing.
- Classify critical points:
- If the derivative changes sign from positive to negative at , then has a local maximum at .
- If the derivative changes sign from negative to positive at , then has a local minimum at .
- If the derivative does not change signs, is neither a local minimum nor maximum.
Example:
Let’s say we have a function .
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Find the derivative:
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Find critical points: Set : So, and are the critical points.
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Determine intervals around the critical points: The critical points divide the number line into three intervals: , , and .
- For , choose : . Since , the function is increasing in this interval.
- For , choose : . Since , the function is decreasing in this interval.
- For , choose : . Since , the function is increasing in this interval.
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Classify critical points:
- At , the function changes from increasing to decreasing, so it has a local maximum.
- At , the function changes from decreasing to increasing, so it has a local minimum.
Conclusion:
- The function is increasing on and .
- The function is decreasing on .
- is a local maximum and is a local minimum.
Would you like to go through another example or discuss any specific part of this process in more detail?
Here are a few related questions:
- How do you apply the second derivative test to classify critical points?
- What happens when the derivative is undefined at a critical point?
- How can you determine if a function is concave up or concave down?
- What are inflection points, and how do they relate to the concavity of a function?
- How can you use a sign chart to analyze a function’s behavior?
Tip: When finding critical points, always double-check your derivative for errors before classifying intervals. It's easy to overlook mistakes in the process!
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Math Problem Analysis
Mathematical Concepts
Derivatives
Critical Points
Intervals of Increase and Decrease
Local Minima and Maxima
Formulas
f'(x) = 0 for critical points
Test for increasing/decreasing intervals (f'(x) > 0 for increasing, f'(x) < 0 for decreasing)
Theorems
First Derivative Test for Local Extrema
Suitable Grade Level
Grades 11-12
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